Remark 2 The transpose of a tensor
\(\mathcal {A}\in \mathbb {R}^{I_1\times \ldots \times I_N\times J_1\times \ldots \times J_M}\)  is a tensor
\(\mathcal {B}\in \mathbb {R}^{J_1\times \ldots \times J_M\times I_1\times \ldots \times I_N}\)  such that
\(\mathcal {A}_{i_1,\ldots ,i_N,j_1,\ldots ,j_M}=\mathcal {B}_{j_1,\ldots ,j_M,i_1,\ldots ,i_N}\) ; see, e.g.,
[1]}. A well-known tensor-matrix product is the so-called \(n\) -mode product
[2]}, which is the same as the Einstein product when summing over the last mode of
the tensor. In other words, we have
\(\mathcal {A}\times _{N+1}A^T = \mathcal {A}\ast _1A,\) 
